Discretization of Partial Differential Equations

Abstract

Analytical solutions of partial differential equations involve closed-form expressions which give the variation of the dependent variables continuously throughout the domain. In contrast, numerical solutions can give answers at only discrete points in the domain, called grid points. For example, consider Figure 5.1, which shows a section of a discrete grid in the xy-plane. For convenience, let us assume that the spacing of the grid points in the x-direction is uniform, and given by ∆x, and that the spacing of the points in the y-direction is also uniform, and given by ∆y, as shown in Figure 5.1. In general, ∆x and ∆y are different. Indeed, it is not absolutely necessary that ∆x or ∆y be uniform; we could deal with totally unequal spacing in both directions, where ∆x is a different value between each successive pairs of grid points, and similarly for ∆y. However, the vast majority of CFD applications involve numerical solutions on a grid which involves uniform spacing in each direction, because this greatly simplifies the programming of the solution, saves storage space and usually results in greater accuracy. This uniform spacing does not have to occur in the physical xy space; as is frequently done in CFD, the numerical calculations are carried out in a transformed computational space which has uniform spacing in the transformed independent variables, but which corresponds to non-uniform spacing in the physical plane. These matters will be discussed in detail in Chapter 6. In any event, in this chapter we will asume uniform spacing in each coordinate direction, but not necessarily equal spacing for both directions, i.e. we will assume ∆x and ∆y to be constants, but that ∆x does not have to equal ∆y.